Circle Calculator

**Circle formulas:** Given: - r = radius - d = diameter = 2r - C = circumference - A = area - π ≈ 3.14159 Relationships: - d = 2r - C = 2πr = πd - A = πr² = π(d/2)² = (πd²)/4 **Inverse formulas (find r from known):** - From diameter: r = d/2 - From circumference: r = C/(2π) - From area: r = √(A/π) **Worked examples:** **Given radius r = 5:** d = 10 C = 2π × 5 ≈ 31.42 A = π × 25 ≈ 78.54 **Given circumference C = 31.42:** r = 31.42 / (2π) ≈ 5 d = 10 A ≈ 78.54 **Given area A = 78.54:** r = √(78.54/π) ≈ 5 d = 10 C ≈ 31.42 All forms give consistent values. **Pi (π) facts:** - π ≈ 3.14159265358979323846... - π is irrational (non-repeating decimal). - π is transcendental (not root of any polynomial). - Discovered ancient: Babylonians ≈ 3.125, Egyptians ≈ 3.16. - Computed: trillions of digits using BBP formula. - For practical work: 3.14 or 3.14159 sufficient. **Common circle measurements:** | r | d | C | A | |---|---|---|---| | 1 | 2 | 6.28 | 3.14 | | 5 | 10 | 31.42 | 78.54 | | 10 | 20 | 62.83 | 314.16 | | 100 | 200 | 628.32 | 31,416 | | 1,000 | 2,000 | 6,283 | 3,141,593 | **Sector and arc:** **Arc length** (portion of circumference at angle θ): - Radians: L = rθ - Degrees: L = (θ/360) × 2πr **Sector area** (pie-slice): - Radians: A_sector = (1/2) × r²θ - Degrees: A_sector = (θ/360) × πr² **Chord** (straight line between two points on circle): chord = 2r × sin(θ/2) **Inscribed/circumscribed:** For polygon inscribed in circle (vertices touch): - Square: side = r√2 - Hexagon: side = r - Triangle: side = r√3 For polygon circumscribed (sides tangent): - Square: side = 2r - Hexagon: side = 2r/√3 **Common circle applications:** - Wheel circumference × revolutions = distance traveled. - Pipe cross-section area = πr² (for flow calculations). - Pulley circumference × RPM = belt speed. - Manhole cover: round so it can't fall through. **Annulus (ring) area:** A = π(R² - r²) Where R = outer radius, r = inner radius. Used for washers, pipe walls, donut shapes. **Ellipse (related shape):** A ellipse with semi-axes a and b: A = πab Perimeter ≈ π[3(a+b) - √((3a+b)(a+3b))] (Ramanujan approximation) When a = b: ellipse becomes circle, formulas reduce. **Polar coordinates:** Circle of radius r centered at origin: x = r cos(θ) y = r sin(θ) Or in Cartesian: x² + y² = r². **Sphere (3D circle):** For a sphere with radius r: - Surface area: 4πr² - Volume: (4/3)πr³ **Cylinder:** For cylinder with circular cross-section, radius r, height h: - Volume: πr²h - Lateral surface area: 2πrh - Total surface area: 2πr² + 2πrh = 2πr(r+h) **Cone:** For cone with base radius r, height h, slant l = √(r² + h²): - Volume: (1/3)πr²h - Lateral surface: πrl - Total surface: πr² + πrl **Unit conversions:** | From | To | Factor | |---|---|---| | cm² | m² | 10⁻⁴ | | in² | cm² | 6.452 | | ft² | m² | 0.0929 | | in | cm | 2.54 | **Practical examples:** | Object | r | A | C | |---|---|---|---| | 1 cent coin (US) | 9.5 mm | 284 mm² | 60 mm | | Soda can (top) | 33 mm | 3,421 mm² | 207 mm | | Tire (typical car) | 35 cm | 3,848 cm² | 220 cm (per rev) | | Manhole cover | 30 cm | 2,827 cm² | 188 cm | | 8-inch pizza | 4 in | 50 in² | 25 in | | Olympic track inner | 36.5 m | 4,185 m² | 229 m | | Earth | 6,371 km | 1.275 × 10⁸ km² | 40,030 km | **Circle inscribed in square:** Circle in square of side s: - r = s/2. - Area ratio = πr²/s² = π/4 ≈ 0.785. - ~21.5% of square is outside circle. **Square inscribed in circle:** Square inside circle of radius r: - Diagonal of square = diameter = 2r. - Side of square = 2r/√2 = r√2. - Area ratio = (r√2)²/πr² = 2/π ≈ 0.637. - ~36% of circle is outside square. **Circle vs square (same perimeter):** For perimeter P: - Square: side P/4, area P²/16. - Circle: r = P/(2π), area P²/(4π). Ratio: circle area / square area = π/4 / (1/4)... Actually: circle area / square area = (P²/(4π)) / (P²/16) = 16/(4π) = 4/π ≈ 1.273. Circle has ~27% more area than square with same perimeter. This is why circular containers are most efficient. **Common applications:** - **Engineering**: pipe sizing, gear design, pulley systems. - **Architecture**: domes, arches, columns, rotundas. - **Construction**: round patios, foundations, swimming pools. - **Manufacturing**: round parts, container design. - **Cooking**: pizza, pies, round cake pans. - **Sports**: track designs, throwing circles. - **Astronomy**: planet sizes, orbital paths. - **Optics**: lens curvature, apertures.